This paper presents the design and simulation of an energy-based control strategy to stabilize a quadrotor described with unit quaternions and their axis-angle representation. The mathematical model is based on Euler-Lagrange formulation using a logarithmic mapping in the quaternion space. The proposed solution introduces a new approach: a quaternion-energy-based control, which introduces an energy expression defined as a Lyapunov function. The designed control law allows the stabilization of the vehicle in all its states. Simulation results and an experimental validation are presented to corroborate the effectiveness of the proposed strategy.